Abstract
The article surveys ideas emerging within the predicative tradition in the foundations of mathematics, and attempts a reading of predicativity constraints as highlighting different levels of understanding in mathematics. A connection is made with two kinds of error which appear in mathematics: local and foundational errors. The suggestion is that ideas originating in the predicativity debate as a reply to foundational errors are now having profound influence to the way we try to address the issue of local errors. Here fundamental new interactions between computer science and mathematics emerge.
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Notes
- 1.
Dana Scott in his opening talk at the The Vienna Summer of Logic (9th–24th July 2014) suggested that we are now witnessing a paradigm change in logic and mathematics. At least in certain areas of mathematics, there is an urgent need to solve complex and large proofs, and this requires computers and logic to work together to make progress. See Dana Scott’s e-mail to the Foundations of Mathematics mailing list of 28–07–14 (http://www.cs.nyu.edu/mailman/listinfo/fom).
- 2.
- 3.
See [3] for a rich discussion of the impact of the paradoxes on mathematical logic.
- 4.
According to a notion of predicativity given the natural numbers which is discussed in the next section.
- 5.
See [6] for an informal account of this notion of predicativity and for further references.
- 6.
The resulting notion of predicativity is, in fact, more generous than in Weyl’s original proposal. The proof theoretic strength of a modern version of Weyl’s system, like, for example, Feferman’s system W from [5], equates that of Peano Arithmetic, and thus lays well below \(\varGamma _0\).
- 7.
This line of research has been brought forward with Feferman’s notion of unfolding, as analysed further by Feferman and Strahm e.g. in [7].
- 8.
As such, Nelson’s ideas have proved extremely fruitful, as they have paved the way for substantial contributions to the area of computational complexity [2].
- 9.
- 10.
A view along similar lines is also hinted at by Feferman in [6].
- 11.
Further challenges are also posed by technical developments in proof theory which have brought Gerhard Jäger to introduce a notion of metapredicative [9]. A thorough analysis of predicativity also ought to clarify its relation with metapredicativity.
- 12.
Predicativity is also at the centre of Martin-Löf’s meaning explanations for type theory, which explain the type theoretic constructions of this theory “from the bottom up”. A key concept here is that of evidence: constructive type theory represents a form of mathematics which is, according to its proponents, intuitively evident, amenable to contentual and computational understanding. This contentual understanding is then seen as supporting the belief in the consistency of this form of mathematics [13].
- 13.
The calculus of constructions takes an opposite route compared with Martin-Löf type theory to the impasse given by Girard’s paradox: it relinquishes the Curry–Howard hisomorphism in favour of impredicative type constructions. Although the Coq sytem was originally developed on the impredicative calculus of constructions, recent versions are based on a predicative core, although they also allow for impredicative extensions.
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Acknowledgements
The author would like to thank Andrea Cantini and Robbie Williams for reading a draft of this article. She also gratefully acknowledges funding from the School of Philosophy, Religion and History of Science, University of Leeds.
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Crosilla, L. (2015). Error and Predicativity. In: Beckmann, A., Mitrana, V., Soskova, M. (eds) Evolving Computability. CiE 2015. Lecture Notes in Computer Science(), vol 9136. Springer, Cham. https://doi.org/10.1007/978-3-319-20028-6_2
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